Algebraic phase unwrapping for functional data analytic estimations—Extensions and stabilizations

The phase unwrapping, which is a problem to reconstruct the continuous phase function of an unknown complex function from its finite observed samples, has been a key for estimating useful physical quantity in many signal and image processing applications. In the light of the functional data analysis, it is natural to estimate first the unknown complex function by a certain piecewise complex polynomial and then to compute the exact unwrapped phase of the piecewise complex polynomial with the algebraic phase unwrapping algorithms. In this paper, we propose several useful extensions and numerical stabilization of the algebraic phase unwrapping along the real axis. The proposed extensions include (i) removal of a certain critical assumption premised in the original algebraic phase unwrapping, and (ii) algebraic phase unwrapping for a pair of bivariate polynomials. Moreover, in order to resolve certain numerical instabilities caused by the coefficient growth in an inductive step in the original algorithm, we propose to compute directly a certain subresultant sequence without passing through the inductive step.